YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS

1999 ◽  
Vol 01 (01) ◽  
pp. 1-50 ◽  
Author(s):  
YANYAN LI ◽  
MEIJUN ZHU
Keyword(s):  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Dimensional Sphere ◽  
Compact Set ◽  
Curvature Function ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Necessary And Sufficient

A theorem of Escobar and Schoen asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, a necessary and sufficient condition for a C2 function K to be the scalar curvature function of some conformal metric is that K is positive somewhere. We show that for any positive C2 function K, all such metrics stay in a compact set with respect to C3 norms and the total Leray-Schauder degree of all solutions is equal to -1. Such existence and compactness results no longer hold in such generality in higher dimensions or on manifolds conformally equivalent to standard three dimensional spheres. The results are also established for more general Yamabe type equations on three dimensional manifolds.

scholarly journals On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1941
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Haila Alodan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Rigidity Results ◽  
Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

scholarly journals On the forces that cable webs under tension can support and how to design cable webs to channel stresses

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180781 ◽  
Author(s):  
Guy Bouchitté ◽  
Ornella Mattei ◽  
Graeme W. Milton ◽  
Pierre Seppecher
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Structural Engineering ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In many applications of structural engineering, the following question arises: given a set of forces f 1 ,  f 2 , …,  f N applied at prescribed points x 1 ,  x 2 , …,  x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 ,  x 2 , …,  x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 ,  f 2 , …,  f N applied at points strictly within the convex hull of x 1 ,  x 2 , …,  x N . In three dimensions, we show that, by slightly perturbing f 1 ,  f 2 , …,  f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

scholarly journals Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1429-1444 ◽  
Author(s):  
Cengizhan Murathan ◽  
Erken Küpeli
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Submersion ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Cosymplectic Manifolds ◽  
Necessary And Sufficient ◽  
Decomposition Theorems ◽  
Characteristic Vector Field

We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ? is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.

scholarly journals Bracing Zonohedra With Special Faces

2015 ◽  
Vol 3 (1-2) ◽  
pp. 88-95 ◽  
Author(s):  
Gyula Nagy
Keyword(s):  
Convex Polyhedron ◽  
Three Dimensional ◽  
Equivalence Classes ◽  
Frame Structure ◽  
Sufficient Condition ◽  
Preliminary Design ◽  
Necessary And Sufficient Condition ◽  
Special Assumption ◽  
Centrally Symmetric ◽  
Necessary And Sufficient

Abstract The analysis of simpler preliminary design gives useful input for more complicated three-dimensional building frame structure. A zonohedron, as a preliminary structure of design, is a convex polyhedron for which each face possesses central symmetry. We considered zonohedron as a special framework with the special assumption that the polygonal faces can be deformed in such a way that faces remain planar and centrally symmetric, moreover the length of all edges remains unchanged. Introducing some diagonal braces we got a new mechanism. This paper deals with the flexibility of this kind of mechanisms, and investigates the rigidity of the braced framework. The flexibility of the framework can be characterized by some vectors, which represent equivalence classes of the edges. A necessary and sufficient condition for the rigidity of the braced rhombic face zonohedra is posed. A real mechanical construction, based on two simple elements, provides a CAD prototype of these new mechanisms.

scholarly journals Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces

2018 ◽  
Vol 6 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Riemannian Manifolds ◽  
Short Note ◽  
Sufficient Condition ◽  
Metric Measure Spaces ◽  
Necessary And Sufficient Condition ◽  
Local Dimension ◽  
Base Points ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient

Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

scholarly journals Some remarks on the spaceR2(E)

1983 ◽  
Vol 6 (3) ◽  
pp. 459-466
Author(s):  
Claes Fernström
Keyword(s):  
Compact Subset ◽  
Complex Plane ◽  
Rational Functions ◽  
Compact Set ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Point Evaluation ◽  
Bounded Point Evaluation ◽  
Point Evaluations ◽  
Necessary And Sufficient

LetEbe a compact subset of the complex plane. We denote byR(E)the algebra consisting of the rational functions with poles offE. The closure ofR(E)inLp(E),1≤p<∞, is denoted byRp(E). In this paper we consider the casep=2. In section 2 we introduce the notion of weak bounded point evaluation of orderβand identify the existence of a weak bounded point evaluation of orderβ,β>1, as a necessary and sufficient condition forR2(E)≠L2(E). We also construct a compact setEsuch thatR2(E)has an isolated bounded point evaluation. In section 3 we examine the smoothness properties of functions inR2(E)at those points which admit bounded point evaluations.

Oscillation and Global Attractivity in a Periodic Delay Equation

1996 ◽  
Vol 39 (3) ◽  
pp. 275-283 ◽  
Author(s):  
J. R. Graef ◽  
C. Qian ◽  
P. W. Spikes
Keyword(s):  
Global Attractivity ◽  
Positive Periodic Solution ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Periodic Delay ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractConsider the delay differential equationwhere α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

scholarly journals Hemi-slant ξ⊥-Riemannian submersions in contact geometry

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3747-3758
Author(s):  
Ramazan Sari ◽  
Mehmet Akyol
Keyword(s):  
Riemannian Manifolds ◽  
Natural Generalization ◽  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Sasakian Manifolds ◽  
Riemannian Submersions ◽  
Space Forms ◽  
Base Manifold ◽  
Sasakian Space Forms ◽  
Necessary And Sufficient

M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

scholarly journals Some properties of lightlike submanifolds of semi-Riemannian manifolds

2010 ◽  
Vol 43 (3) ◽  
Author(s):  
Rakesh Kumar ◽  
Rachna Rani ◽  
R. K. Nagaich
Keyword(s):  
Riemannian Manifolds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Totally Geodesic ◽  
Screen Distribution ◽  
Necessary And Sufficient ◽  
Lightlike Submanifolds ◽  
Lightlike Submanifold

AbstractWe initially obtain various relations and then establish necessary and sufficient condition for the integrability of screen distribution of a lightlike submanifold. We also establish necessary and sufficient condition for a lightlike submanifold to be totally geodesic.

Oscillations of Second Order Neutral Equations

1988 ◽  
Vol 40 (6) ◽  
pp. 1301-1314 ◽  
Author(s):  
G. Ladas ◽  
E. C. Partheniadis ◽  
Y. G. Sficas
Keyword(s):  
Second Order ◽  
List Type ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Neutral Equations ◽  
Deviating Arguments ◽  
Real Roots ◽  
All Solutions ◽  
Image Position ◽  
Necessary And Sufficient

Consider the second order neutral differential equation1where the coefficients p and q and the deviating arguments τ and σ are real numbers. The characteristic equation of Eq. (1) is2The main result in this paper is the following necessary and sufficient condition for all solutions of Eq. (1) to oscillate.THEOREM. The following statements are equivalent:(a) Every solution of Eq. (1) oscillates.(b) Equation (2) has no real roots.

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